Product Rule Explanation

It is not always necessary to compute
derivatives directly from the definition. Several rules have been developed
for finding the derivatives without having to use the definition directly. These
rules simplify the process of differentiation. The Product Rule is a formula developed
by Leibniz used to find the derivatives of products of functions.

The Product Rule is defined as the product of the first function and the derivative
of the second function plus the product of the derivative of the first function
and the second function:

The Formula for the Product Rule

Product Rule Example

Find f'(x) of

We can see that there is a product, so we can apply the product rule. First, we
take the product of the first term and the derivative of the second term.

Second, we take the product of the derivative of the first term and the second term.

Then, we add them together to get our derivative.

Notice that if we multiplied them together at the start, the product would be 21x5.
Taking the derivative after we multiplied it out would give us the same answer -
105x4. The product rule helps take the derivative of harder products of functions
that require you use the rule instead of multiplying them together beforehand.

Let's look at a harder example:

Differentiate:

We can see that we cannot multiply first and then take the derivative. We must use
the product rule.

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